Note on the spectra of Steiner distance hypermatrices
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| Format: | Preprint |
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2024
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| _version_ | 1866909127402323968 |
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| author | Cooper, Joshua Du, Zhibin |
| author_facet | Cooper, Joshua Du, Zhibin |
| contents | The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \times \cdots \times n$ ($k$ terms) array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of $k=2$, this reduces to the classical distance matrix of a graph. Graham and Pollak showed in 1971 that the determinant of the distance matrix of a tree only depends on its number $n$ of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) $n \geq 3$ and $k$ is odd, (b) $n=1$, or (c) $n=2$ and $k \equiv 1 \pmod{6}$. Two proofs are presented of the $n=2$ case -- the other situations were handled previously -- and we use the argument further to show that the distance spectral radius for $n=2$ is equal to $2^{k-1}-1$. Some related open questions are also discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Note on the spectra of Steiner distance hypermatrices Cooper, Joshua Du, Zhibin Combinatorics 05C12 (Primary) 05C50, 15A69 (Secondary) G.2.2 The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \times \cdots \times n$ ($k$ terms) array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of $k=2$, this reduces to the classical distance matrix of a graph. Graham and Pollak showed in 1971 that the determinant of the distance matrix of a tree only depends on its number $n$ of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) $n \geq 3$ and $k$ is odd, (b) $n=1$, or (c) $n=2$ and $k \equiv 1 \pmod{6}$. Two proofs are presented of the $n=2$ case -- the other situations were handled previously -- and we use the argument further to show that the distance spectral radius for $n=2$ is equal to $2^{k-1}-1$. Some related open questions are also discussed. |
| title | Note on the spectra of Steiner distance hypermatrices |
| topic | Combinatorics 05C12 (Primary) 05C50, 15A69 (Secondary) G.2.2 |
| url | https://arxiv.org/abs/2403.02287 |